Sums of Hermitian Operators and Connections to Connes' Embedding Problem; Hyperinvariant Subspaces

厄米算子之和以及与 Connes 嵌入问题的联系；

• 批准号: 0901220
• 负责人: Kenneth Dykema
• 金额: $ 24.52万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901220&HistoricalAwards=false
• 关键词: Sums; Hermitian; Operators; Connections; Connes; Sums; Hermitian; Operators; Connections; Connes

AbstractDykemaThe PI will investigate two fundamental problems in the theory of operators that are contained in II_1 factors. The first is Connes' embedding problem. Recent work of Collins and Dykema has shown that this problem is equivalent to a question about sums of operators in finite von Neumann algebras, and other recent work of Bercovici, Collins, Dykema, Li and Timotin has positively answered the first part of this question, showing that all Horn inequalities hold in all finite von Neumann algebras. The second problem is the hyperinvariant subspace problem. In particular, the PI will focus on the remaining open part of this problem for elements of II_1-factors, namely, the case of quasi-nilpotent operators in II_1-factors.Operators on infinite dimensional Hilbert space are used in mathematical models of quantum mechanics, and they are of significance in diverse areas of mathematics. We will work on two fundamental problems in operator theory: Connes' embedding problem and the hyperinvariant subspace problem. These concern different aspects of the structure of operators on infinite dimensional Hilbert spaces. We will focus on operators whose algebras possess traces. The first problem is about how well such operators can be approximated (in their mixed moments with respect to the trace) by operators on finite dimensional spaces. We will attack this problem by examining eigenvalues of sums of operators. The second problem is about the possibility of decomposing operators on infinite dimensional space by restricting them to invariant subspaces. In some recent progress, Haagerup and Schultz have proved the existence of such subspaces for a large class of operators, and we will focus on some specific operators for which this question is unresolved.

Abstractdykemathe Pi将研究运算符理论中的两个基本问题，其中包含II_1因素。 首先是Connes的嵌入问题。 Collins和Dykema的最新工作表明，这个问题等于有关有限的von Neumann代数中运营商总和的问题，以及Bercovici，Collins，Dykema，Li和Timotin的其他最新工作，都积极回答了这个问题的第一部分，表明所有Horn Equarities在所有有限的Vone Neumann neumann algebras中都持有所有角度。 第二个问题是Hypervariant子空间问题。 特别是，PI将专注于II_1因子元素的剩余部分，即II_1-FACTORS中的Quasi-nilpotent运算符的情况。无限维度希尔伯特空间的操作器用于量子力学数学模型，并且它们在多种多样的数学领域具有重要意义。 我们将研究操作者理论中的两个基本问题：Connes的嵌入问题和Hypervariant子空间问题。 这些涉及在无限尺寸希尔伯特空间上运营商结构的不同方面。 我们将专注于代数具有痕迹的操作员。 第一个问题是关于操作员在有限的维空间上如何近似（在相对于痕迹的混合矩相对于痕迹的混合力矩）如何近似。 我们将通过检查运营商总和的特征值来攻击这个问题。 第二个问题是关于将操作员限制为不变子空间的可能性。 在最近的某些进展中，Haagerup和Schultz证明了大量运营商的这种子空间的存在，我们将重点关注一些无法解决的特定运营商。